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Yutaka Taniyama

Assuming Taniyama-Shimura. I promised midweek to explain this phrase a little bit more. Now the weekend is here, let's look at a phrase that gave mathematicians sleepless nights in the second half of the twentieth century, and why it is relevant to the coverage and potential product launch of the Apple's third iPad.

The Taniyama-Shimura Conjecture was theorised in 1955 by Yutaka Taniyama and Goro Shimura, and in plain English stated that "every elliptic equation is associated with a modular form." Extended in 1967 by Andre Weil and partially solved for semi-stable elliptic curves in 1995 by Andrew Wiles and Richard Taylor (incidentally solving Fermat's Last Theorem in the process), a strict and rigorous mathematical proof would not be found until 2001. In that 46 year gap, mathematics built up theory, solutions, and explored countless areas of mathematics. And every paper started with a little note along the lines of "...assuming Taniyama-Shimura is correct."

Nowadays there is this solid foundation for mathematics, but for years it could have been wiped out because people went with the emotional 'it looks right' of the relationship between elliptic curves and modular form and not stopping to demand proof.

So what does this mean for the world's favourite consumer experience company based in Cupertino? Seriously, have you seen the amount of coverage the iPad 3 is getting? And yes, that includes this article, but I refer you to the opening paragraph.

Countless words have been written, speculation spoken, and views put down on video over a product that does not exist in public. Where are the sources? We're all expecting it, it emotionally feels right, it fits into a launch pattern from previous years, and there are individual points of reference that could match up (such as MacRumors discovering a 9.7 inch retina screen), and entire websites and careers are built upon the shaky foundation of educated guesses, half truths, and emotion. That's no different to Taniyama recognising the pattern of numbers from an individual Elliptic curve (an E-Series) matching up with the M-Series of a modular form. Just because one piece of data fits, it cannot be extrapolated out to the full product. Not without proof.

And we've no idea when that proof will be announced (but everyone is pretty sure it's March 7th).

Of course for the reporters commenting on any new technology, especially from Apple, there's very little comeback. If a mathematician had come along and disproved Taniyama-Shimura, then almost forty years of number theory could have been wiped out in the blink of an eye. What happens if the iPad 3 has a different screen size to what MacRumors we expect? People just think "ah, that's Apple for you" and the Apple PR team sit back and do their best impression of Montgomery Burns.

So we can all write and speculate, safe in the knowledge that no matter what turns up, no matter if we are proved right, no matter if another writer lands the coveted 'first!' story, we all know at least one simple fact.